Monday, February 09, 2009

Confirmatory and disconfirmatory evidence

Much is made of the "logical fallacy" of evidentiary reasoning. The hypothetico-deductive method supposedly goes like this:

    (A) If theory X were true, we should see Y; we see Y, therefore theory X is true
The problem is obvious: it's the fallacy of the affirmation of the consequent. In logic, the converse of an implication cannot be directly derived an implication: if p -> q is true, we cannot always infer that q -> p is true.

We have a better tool, though, in our toolbox of logic: the contrapositive:

    (B) If theory X were false, we should not see Y; we do see Y, therefore theory X is true
This has the advantage of logical validity: if ~p -> ~q, we can validly infer that q -> p.

We want to make sure that theory X does in fact preclude something. Consider the statements about a "toy" theory of gravity:
    (1) If the theory of gravity is true, a dropped rock will always move somewhere at some acceleration
    (2) If the theory of gravity is true, a dropped rock will always accelerate straight down at 10 m/s2
Both statements are of form of an implication (p -> q). We cannot, however, create a negative implication from the first implication: the "theory" predicts any logically possible observation of its motion. On the other hand, we can easily create a negative implication from the second:

    (2') If the theory of gravity were false, a dropped rock will sometimes not accelerate straight down at 10 m/s2
Theory (2), therefore, is falsifiable.

The presence of "always" and "sometimes", however, complicates the matter. Life would be easier if I could say
    (2''): If the theory of gravity were false, a dropped rock will never accelerate straight down at 10 m/s2
Unfortunately, we cannot derive this implication from (2). If, therefore, I drop a rock and it accelerates straight down at 10 m/s2, I have not demonstrated that the consequent is false, and I cannot employ the contrapositive to infer the truth of the antecedent.

We can use probabilism, however, to quantify "sometimes". We can say,
    (2p) If the theory of gravity were false, the position of a dropped rock at time t should be randomly distributed (or correlated with some point other than p)
We can calculate the probability of the rock randomly being at point p at time t (straight down after accelerating at 10 m/s2). We can then use Bayes' Theorem to calculate the probability that the rock always appears at point p after time t.

The "metaphysical" objection to Bayes' theorem is that it requires the assumption of some specific prior probability that our theory is true, and we have no good prior reason to assign any that specific probability. There are two ways of overcoming this objection. First, we can simply set the prior probability for the truth of the theory to the probability that the rock will appear randomly at point p at time t. Our first (successful) trial, then, will raise the probability that the theorem is true to 50% (i.e. perfectly agnostic).

Another strategy is to plug any arbitrary value (such as 50%) as the prior probability. Each subsequent successful experiment will raise the probability that the theorem is true; if the theory really is true, one can perform a finite number of experiments to obtain any desired probability, and as the desired probability rises exponentially, the number of experiments necessary to achieve that probability increases linearly.

It's important to understand that the "truth" of a theory is, in this sense, nothing more (or less) than the prediction of a correlation. This interpretation of theoretical truth ignores any underlying ontological interpretation of a theory, and ignore the the sense of "truth" as that the ontological interpretation "corresponds to" reality. A "true" theory in this sense is a theory that predicts or explains correlations.

Much philosophical hay has been made of the undesirability of making ad hoc corrections to a theory. But that a theory is adjusted on the basis of evidence that contradicts the theory is not by itself objectionable. We have to look, rather, at whether ad hoc corrections to a theory make it broader or narrower, predicting more or fewer alternative observations. For example, Einstein's "ad hoc" corrections to Newton's theory of gravity is just as narrow, just as specific, as Newton's theory.

With this way of looking at evidentiary arguments, we can easily see many of the problems with evidentiary apologetics. The biggest problem with evidentiary apologetics is that they do not specify disconfirmatory evidence, except in the trivial sense that we would not have the specific confirmatory evidence we already have.

Another more subtle problem with evidentiary apologetics is that by the invocation of supernaturalism they deny the probabilistic basis of determining universal physical laws. Supernaturalism essentially says that even if an observation precluded by universal physical law is actually observed, the "universal" physical law might still be true. According to the natural theory of gravity, the probability of it rising straight up is exactly 0% — if I drop a rock and it rises, my theory is decisively disproven. Under supernaturalism, though, the probability that it will "miraculously" rise straight up is not calculable. No series of observations, however many we perform, can ever support or undermine an assertion of universal physical law: We can attribute any number of contrary observations to miracles.

William Lane Craig, for example, invokes supernaturalism.
What, after all, is the resurrection hypothesis? It’s the hypothesis that Jesus rose supernaturally from the dead. It is not the hypothesis that Jesus rose naturally from the dead. That Jesus rose naturally from the dead is fantastically improbable. But I see no reason whatsoever to think that it is improbable that God raised Jesus from the dead.
The problem is obvious: There is no way to calculate the prior probability of a supernatural event. He makes this interpretation clear in the preceding sentence:
In order to show that that hypothesis is improbable, you’d have to show that God’s existence is improbable. But Dr. Ehrman says that the historian cannot say anything about God. Therefore, he cannot say that God’s existence is improbable. But if he can’t say that, neither can he say that the resurrection of Jesus is improbable. So Dr. Ehrman’s position is literally self-refuting.
Craig is right: We simply cannot discuss the prior probability of God's existence and therefore of a supernatural resurrection. There is no meaningful way — not even a metaphorical way — of discussing the prior probability of a supernatural event. How, for example, can we speak even metaphorically about the "possible worlds" where God does or does not resurrect Jesus?

Saying that a number in an equation is not meaningful does not give us license to plug in whatever value we feel like. To say that the prior probability of Jesus' resurrection is not calculable means that one cannot draw any inferences at all from a Bayesian examination of the evidence.

This is why I assert that Craig is at best (as a professional academic) massively incompetent or at worst intentionally lying. He proceeds with an argument he ought to know is logically invalid. If he does not know it's invalid, he's incompetent; if he knows it's invalid and still uses it, he's lying. Furthermore, it is very difficult to believe that everyone in the field of evidentiary apologetics is massively incompetent: it seems highly probable that some of them know that the argument from evidence is invalid because of supernatural priors, but still continue to employ — and allow their incompetent colleagues to employ — the argument.

In either case, it is pointless to engage in a rational examination of evidence with evidentiary apologists. They are either incompetent to employ the argument, or they're lying.

1 comment:

  1. Following Craig's example, this should work as well:
    What, after all, is the transmission hypothesis? It’s the hypothesis that the angel Gabriel supernaturally transmitted the Qur'an to the Prophet. It is not the hypothesis that the Qur'an was obtained naturally. That the perfection of the Qur'an was obtained naturally is fantastically improbable. But I see no reason whatsoever to think that it is improbable that God prompted Gabriel to read the Qur'an to the Prophet.

    As long as supernatural (i.e., anything goes) explanations are allowed, I could also rewrite that paragraph to feature Joseph Smith and the angel Moroni; or practically anything that even remotely correlates with other bits and pieces of historical evidence, really.

    I suppose this is one of the reasons why religions tend to continue to grow through fragmentation, rather than merging: asking which collection of myths is "factual" is a category mistake.

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